Energy distribution and economic growth§

Carl-Johan Dalgaard a, Holger Strulik b,*a University of Copenhagen, Department of Economics, Studiestraede 6, 1455 Copenhagen K, Denmarkb University of Hannover, Wirtschaftswissenschaftliche Fakultat, Konigsworther Platz 1, 30167 Hannover, Germany

Resource and Energy Economics 33 (2011) 782–797

A R T I C L E I N F O

Article history:

Received 30 September 2008

Received in revised form 10 June 2010

Accepted 27 April 2011

Available online 27 May 2011

JEL classification:

O11

O13

Q43

Keywords:

Economic growth

Energy

Power laws

Networks

A B S T R A C T

This research examines the physical constraints on the growth

process. In order to run, maintain and build capital energy is

required to be distributed to geographically dispersed sites where

investments are deemed profitable. We capture this aspect of

physical reality by a network theory of electricity distribution. The

model leads to a supply relation according to which feasible

electricity consumption per capita rises with the size of the

economy, as measured by capital per capita. Specifically, the

relation is a simple power law with an exponent assigned to capital

that is bounded between 1/2 and 3/4, depending on the efficiency of

the network. Together with an energy conservation equation,

capturing instantaneous aggregate demand for electricity, we are

able to provide a metabolic-energetic founded law of motion for

capital per capita that is mathematically isomorphic to the one

emanating from the Solow growth model. Using data for the 50 US

states 1960–2000, we examine the determination of growth in

electricity consumption per capita and test the model structurally.

The model fits the data well. The exponent in the power law

connecting capital and electricity is 2/3.

� 2011 Elsevier B.V. All rights reserved.

§ We owe a special thanks to Jayanth Banavar for generously sharing his expertise on the subject of energy distributing

networks. We also thank Michael Burda, Henrik Hansen, Martin Kaae Jensen, seminar participants at the Universities of

Birmingham, Copenhagen, Gothenburg, Hannover, Humboldt University Berlin and the 2008 SURED Conference, two

anonymous referees and the editor, Sjack Smulders, for helpful comments.* Corresponding author.

E-mail addresses: carl.johan.dalgaard@econ.ku.dk (C.-J. Dalgaard), strulik@vwl.uni-hannover.de, holger_strulik@brown.edu

(H. Strulik).

Contents lists available at ScienceDirect

Resource and Energy Economics

jo u rn al h om epag e: ww w.els evier .c o m/lo c at e/ ree

0928-7655/$ – see front matter � 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.reseneeco.2011.04.004

1. Introduction

The work of Domar (1946) and Solow (1956) marked the beginning of the formal analysis of thegrowth process where physical capital accumulation is seen a key growth engine. This body ofresearch has unveiled fundamental structural characteristics which impinge on the determination oflabor productivity in the long run: savings, population growth, technological change and more. Thesefactors share the common feature that they importantly affect the ability of an economy to mobilizeresources for the purpose of capital accumulation.

At the same time, the basic neoclassical growth theory abstracts from the fact that it makes littlesense to acquire a piece of machinery, at a particular time and place, unless the machine can besupplied with electricity and put to use. Surely, it is of first order importance that an economy is able todistribute electricity across the economy to the sites where investments are deemed profitable.1

Omitting the physical preconditions for growth may be highly problematic as such factors canrepresent a binding constraint on economic growth. Hence, in the present study we provide anattempt to model electricity distribution, and take a first step towards examining its implications forthe growth process.2

Electricity networks are highly complex systems; too complex, one might think, for macroeconomicmodeling. Fortunately, progress has been made in the natural sciences in describing and modeling theaggregate properties of similarly complex networks. The work of West et al. (1997, 1999) and Banavaret al. (1999, 2002) is a case in point. By modeling biological organisms as an energy distributing networksthese authors have been able to show, in keeping with the evidence, why the energy needs of anorganism as a whole rises with the size of the organism in accordance with B=B0 �mb, where B is basalmetabolism, m is body mass, B0 is a constant, and b=3/4.3 In the present context one might wonder if asimilar sort of result could be derived in the case of an electricity network, thus linking electric powerconsumption and a measure of the ‘‘size’’ of an economy, such as the capital stock.

At first it may seem far-fetched to believe that empirical laws and mathematical theoriespertaining to biological organisms should have any sort of bearing on man-made networks. But uponreflection the link is perhaps not improbable, for three reasons.

First, the cardiovascular system and the power grid share the feature of being energy distributingnetworks (of nutrients in the former case, electricity in the latter). Conceptually they are thereforehighly related. Of course, the networks are very different at the more detailed level: in appearance andin terms of the matter being distributed. Nevertheless, and in spite of being developed for theinvestigation of biological systems, the inventors of the network theory that we employ to the study ofelectricity distribution below suggest themselves that their theory could be adapted to the case ofelectrical currents (Banavar et al., 1999, p. 132).

Second, there is a good reason why biological networks and man-made networks would come tohave similar aggregate properties. Both biological and man-made networks are developed over timethrough a process of gradual optimization. In the former case this occurs through natural selection; inthe latter it is the result of deliberate decisions to rework, extend and improve the efficiency of thenetwork in question. Undoubtedly, the process of natural selection has worked to produce efficientnetworks. Quite possibly, man-made networks have moved in a similar direction. As efficientnetworks have certain unique properties (e.g., minimal transmission losses), biological and man-madenetworks should have some aggregate characteristics in common.

1 Empirically, the close link between electricity and growth is well documented. For instance, in Henderson et al. (2009) the

authors propose to use satellite data on lights at night as an alternative proxy for GDP.2 The present analysis is therefore related to the literature on infrastructure and economic growth. The traditional approach

essentially consists of adding another input into the production function, thus capturing infrastructure capital (e.g., Arrow and

Kurz, 1970 and many others since). Our approach to model the influence from the electricity infrastructure will differ in that we

will provide a model for the network itself, and derive the critical associations between electricity use and capital. At the same

time we abstract from other dimensions of infrastructure, like roads, ports, etc. Also, we abstract from the problems associated

with the production of electricity, which in practise requires the use of natural resources. See e.g. Stiglitz (1974) and Suzuki

(1976) for a discussion of sustainable growth in the presence of exhaustible natural resources.3 This formula is known as ‘‘Kleiber’s law’’ (Kleiber, 1932), which, remarkably, holds across biological organisms spanning 27

orders of magnitude in mass; from the molecular level up to whales (West and Brown, 2005).

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797 783

Third, physicists have already started to apply the empirical methods from the study of biologicalorganisms to the case of artificial networks in an effort to uncover universal scaling laws with bearingon human societies (e.g., Bettencourt et al., 2007).4 This research strategy should be seen andappreciated through the lens of the remarks above. Biological networks and man-made networks havesimilar objectives, namely to distribute resources for final application (be it cell maintenance or thecharging of an ipad) as efficiently as possible. Both networks are under constant selective pressurethrough which they edge towards optimality. In the end it would actually be rather surprising ifbiological networks and man-made networks did not have many common characteristics.

Accordingly, in this study we begin by developing a model of an economy viewed as atransportation network for electricity. The model predicts that electricity consumption per capita (e)can, loosely speaking, be seen as the economic counterpart to metabolism, and capital per capita (k) asthe counterpart to body size; the association between electricity and capital is concave, and log-linearas Kleiber’s law. The relevant interpretation of this power law association is as a supply relation: itcaptures the ability of an economy to make electricity available at geographically dispersed sites forthe purpose of final use.

The model delivers the above mentioned power law association between e and k and offerspredictions regarding the size of the key elasticity linking capital and electricity consumption.Specifically, it is demonstrated that depending on the efficiency of the economy in the context ofelectricity distribution (in a sense to be made precise below), the elasticity should fall in a boundedinterval ranging from 1/2 to 3/4; the more efficient the economy the larger the elasticity. Byimplication, economies that are more efficient in electricity distribution will be able to make moreelectricity available for final use. Ceteris paribus, such economies should be able to accumulate capitalat a faster rate than less efficient economies.

Needless to say, our characterization of electricity distribution neglects a lot of the details of actualpower supply. It remains a crude aggregate representation of the network, for which we can providesome micro foundations. In this sense the equation bears some similarity to the aggregate productionfunction. When macroeconomists write down a production function it is not because they believe thisto be an accurate description of the full complexity of the myriads of production processes thatultimately generate GDP. It is viewed as a useful short cut, which is justified by the need to gain someunderstanding of how the economy operates. Our network representation should be viewed in asimilar light, and is subject to similar limitations.

With an equation summarizing power distribution in hand we subsequently add a representationof electricity demand. From an accounting perspective electricity can be viewed as being used forthree basic purposes: running and maintaining existing capital and creating new capital. The notionof capital is broad, including both electricity consuming capital used by households (e.g., an airconditioner in a living room) as well as capital used by firms (e.g., an air conditioner in a factoryhall).5 If a ‘‘characteristic’’ machine consumes a certain amount of electricity in use and requires acertain amount of electricity to be created, total electricity demand (at an instant in time) is simplythe sum of the electricity requirements related to use, maintenance, and construction of capital.Assuming the demand for electricity (thus determined) equals supply (as reflected in the power lawassociation) we can derive a simple first order differential equation governing the evolution ofcapital per capita over time. We can then proceed to study the implied dynamics and characterizethe steady-state.

The law of motion for capital is mathematically isomorphic to the one emanating from a Solow(1956) model, where the aggregate production function is assumed to be Cobb–Douglas. But in

4 ‘‘Allometric scaling’’ is a technique used in biology to study how selected biological variables of an organism correlate with

the size of the organism. A fundamental allometry is the one mentioned in the text, between energy consumption B and body

mass m of a mammal (‘‘Kleiber’s law’’). In Bettencourt et al. (2007) the authors examine the links between various urban

indicators (like, total number of patents, housing, crime and energy consumption) and city size measured by population.5 As a result, our concept of capital is not the same as the national accounts concept, which only classifies an air conditioner as

capital if it is used by firms; if the (same) air conditioner is placed in a household it is classified as a durable consumption good in

national accounts. In many contexts this distinction is important. In present situation, however, it makes no difference to

electricity requirements whether the machine is placed in a home, or in a factory. As a result, we abstract from the distinction in

the formal analysis below.

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797784

contrast to the Solow model the structure developed below holds predictions for the amount of capital(usable in consumption or production) that an economy can sustain in the long run from a physicalperspective. This level of capital is determined by the efficiency of the electricity distributing network,as well as the energy cost associated with running, maintaining, and creating capital.

Below we argue that technological change is the key driver behind changes in this physical limit tocapital accumulation per capita in the long run. Hence, one should not view the steady-state derivedbelow as an absolute boundary for capital but rather as a constraint which continually is beingmodified due to technological change.

From an empirical perspective, however, the law of motion for capital is difficult to confront withdata since our definition of capital necessarily is broader than the national accounts concept (cf.footnote 5). However, we demonstrate below that the central law of motion for capital per capita canalso be restated in terms of electricity consumption per capita, which is a directly observablevariable.

If economies – on average perhaps – are operating at the boundary of physical feasibility, the modelshould be able to match the data on electricity consumption per capita. In order to examine whetherthis is the case or not, we examine the model’s implications using cross-state data for the 50 US States,1960–2000. In terms of electricity consumption per capita the model holds several strong predictions.First, conditional on structural characteristics of an economy (notably population growth) one wouldexpect to see b-convergence in electricity consumption per capita. Second, the model can bestructurally estimated, which allows for the identification of the networks parameter for which wehave a theoretical prior.

Using cross-sectional data on electricity sales for the 50 US states we find strong support for themodel. Over the period 1960–2000 there is marked tendency for conditional b-convergence inelectricity consumption per capita. Moreover, the 95% confidence interval for the networks parameterconforms with the theoretical predictions: 1/2 to nearly 3/4. The point estimate is about 2/3.

The remaining part of the paper proceeds as follows. Section 2 lays out the model of the economy,viewed as an electricity distributing network, and derives the power law association betweenelectricity consumption and capital. We then proceed, in Section 3, to add electricity demand, andderive the law of motion for capital and electricity consumption, respectively. Section 4 discusses themodel’s implications for long-run development and Section 5 contains the empirical analysis. Finally,Section 6 concludes.

2. The economy as a network

In the biological context, the power law (i.e., Kleiber’s law mentioned above) is derived from thenotion of living organisms as energy transporting networks whereby the size of the organism is givenby the number of energy consuming units, i.e. the number of cells that have to be fed.6 By way ofanalogy we view an economic ‘‘organism’’ as an electricity transporting network whereby the sizeultimately is related to the amount of capital which needs to be ‘‘fed’’. This section therefore extendsthe model due to Banavar et al. (1999) so as to analyze an economy distributing electricity.7 SinceBanavar et al. mentioned the potential applicability of the originally biological theory to man-madenetworks, a couple of articles were published taking inspiration from these ideas to understand the‘‘metabolism’’ of cities, regions, or countries and to investigate scaling laws between population sizeand human consumption (see e.g. Kuhnert et al., 2006; Bettencourt et al., 2007; Isalgue et al., 2007;Samaniego and Moses, 2008).

Fundamentally, the purpose of the network is to deliver (non-human) energy to the electricityconsuming units of the economy. Electricity is assumed to originate from a source (a power plant) andis diffused across the economy via a power grid to the sites at which it is used. In keeping with theestablished terminology (Banavar et al., 1999), we refer to each site as ‘‘a transfer site’’; this is whereenergy is converted into work effort. Each transfer site is assumed to be scale-invariant; as thenetwork expands the geometric size of the individual transfer site does not change. A reasonable way

6 West et al. (1997), West and Brown (2005) and Banavar et al. (1999, 2002).7 For other applications of the theory to drainage basins of rivers, see Maritan et al. (2002) and Rinaldo et al. (2006).

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797 785

to think about the transfer sites is as electricity outlets, which arguably fulfill this requirement.8

Moreover, all transfer sites are locally connected, and thus linked to the source either directly orindirectly via transmission lines.

The size of the network is defined by the geometric size of the shape which defines its outercontours. In biology the outer contour of the network is tangible – the body. In the present case it isabstract; i.e., the geometric shape which would be able to engulf the network. The fact that thisgeometric shape is not tangible is immaterial for the argument.

Let the linear size of the network be denoted by L so that the total size of the network isproportional to LD, where D is the dimension of the network. Since electricity is distributed withinthree-dimensional systems like cities and through three-dimensional structures like houses, plants,and machines, the notion of D=3 is certainly most appropriate. However, the theory nowhere cruciallyhinges on the numerical specification of dimensionality and we can, in fact, let the data decide aboutthe appropriate notion of dimensionality.9

The distance between a transfer site and the source i is defined by the number of transfer sites onewill have to pass to get from i to the source. A bigger network nests more transfer sites. Hence, giventhe distance convention it is therefore inevitable that the mean distance between the transfer sites andthe source rises as the network becomes larger. However, the specific nature of the network mattersfor how large the increase is, as will be seen momentarily.10

Generally, assuming a space-filling network and scale-invariant transfer sites, the number oftransfer sites must rise with the (geometric) size of the network. As each transfer site uses energy, onemay anticipate an association between the size of the network and the total electricity consumed atthe transfer sites, E. More specifically, we assume that for a given size of the network LD total electricityconsumption is linear in population size P.

E / LD � P: (1)

As a result, a change in per capita electricity consumption requires a proportional change in the size ofthe network, e/LD. That is, per capita electricity consumption, e, is ultimately attributable to thenumber of devices (e.g. television sets, washing machines, computers and so on), which a givenpopulation utilizes. The notion is that every time a new piece of equipment is connected to anelectricity outlet, a new transfer site emerges, and the network expands allowing for more electricityconsumption per capita.

In order to understand the proportionality in (1) it is helpful to recall that the linear correlationbetween population size and total electricity consumption holds for a given size of the network. Theassumption is that doubling the number of persons in a household will double electricity consumptionceteris paribus because twice as many people use the appliances connected to the electricity grid. Forexample, a household’s washing machine is used twice as often than before. The effect originatingfrom a change of the number of persons using the electricity grid is thus fundamentally different fromthe effect that originates from adding new machines and appliances to the grid. For example, ifhouseholds complement their washing machine with a dryer, this will enlarge the size of theelectricity distributing network.

In principle one could argue that the E – P association need not be linear; there could be scaleeffects. For example, new household members could use an existing radio without consuming anyextra electricity (if they are listening at the same time as the original household member). Whetherthe association is indeed linear is thus an empirical question. Recent empirical studies investigatingthis particular question confirmed a linear association between E and P for a cross-section of German

8 The size of electricity outlets are in practise independent of the size of the associated building, and the surrounding network.

They also tend to have the same size across countries; electricity outlets are no bigger in rich countries than in poor countries

(though they surely are more plentiful). As a result, maintaining scale invariance of the transfer site may not be unreasonable in

an economic context.9 Recently, an article has been published that complements our macro theory on the micro-level. Moses et al. (2008)

investigate electricity distribution within micro-processors and argue that these objects could be conceptualized as being 2.5-

dimensional because of the relatively thin third dimension originating from the metal layers of wires on micro chips.10 See Banavar et al. (1999) for sketches of networks.

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797786

cities (Kuhnert et al., 2006) and Chinese urban administrative units (Bettencourt et al., 2007). In theempirical section of this paper we provide additional support using cross-state data for the US.

A key result in Banavar et al.’s network theory is the proof of an association between total flow ofenergy in a network, F, and the size of the network given by F/E �Lx, where x depends on the efficiencyof the network. Specifically, x=1 in directed networks that minimize total energy requirementsneeded to fuel the economy (or the organism in biology) subject to the requirement that all sites areserved. In the most inefficient network, the space-filling spiral, x=D. Inserting (1) we get F/LD+x �P.Notice, by comparison with Eq. (1), that an increasing size of the network implies that a greaterfraction of total energy supply (F) is used to ‘‘fuel’’ the system, as opposed to being available forconsumption at the sites (E). This result reflects the fact that when the size of the network rises theenergy flow per capita (F/P) expands at least in proportion to LD+1, and at most in proportion to L2D.

The basic intuition for this result the following. With a directed network, the average distancebetween the transfer sites and the source is minimized, which implies that the total amount of energyneeded to serve the system (F) of a given size is minimized. By contrast, with the space filling spiral,energy needs to be transported (physically speaking) farther to serve each transfer site in the network.Consequently, the amount of energy needed to serve the system in its totality, at any given instant intime, is larger. Hence, the organization of the network may be more or less efficient, in the sense of thetotal amount of energy needed relative to the size of the network (or the number of appliances thatneeds to be fed with electricity).

Finally, we assume proportionality between the total capital stock and total energy in thesystem F /K. This assumption is thought to capture that capital is nested at the transfer sites and

in the network itself, in the form of the transmission lines that connect the transfer sites. Hencecapital is needed to transfer electricity (and ‘‘hosts’’ electricity in the process) to the sites wherecapital uses energy. Energy conservation in the system at large (at any given instant in time)would then suggest proportionality between the capital stock and the total flow of energy in thesystem.

We impose exact proportionality (unit elasticity) based on the following long-run consideration.Suppose we instead assumed F/K

f, where f may differ from 1. If f<1 this would mean that as the

capital stock gets larger, F/K drops, which implies that capital in the limit can be applied without theuse of energy. This seems like an undesirable property. Conversely, if f>1 it implies that energy can bediffused throughout the system, in the limit, without the use of capital. This does not seem plausibleeither (currently at least). That is not to deny that there could be periods during which F/K rises ordeclines, which could be captured by allowing for a specification such as F/K

f, where f differs from 1.

However, we doubt such a state of affairs could be maintained in the long run. As a result, we opt forthe specification where f=1.

The established associations between the capital stock, size of the network, and electricityconsumption per capita, F/K, F/LD+x �P, e/LD can be summarized as a log-linear association betweenelectricity use per capita and the amount of capital per capita k, determined up to a constant e. Itrepresents the reduced form of the economy as a network:

e ¼ eka; a � D

D þ x; x 2 ½1; D�: (2)

The intuition for the concave scaling association is the following. When K rises, new transfer sitesemerge and the size of the network expands. As a result, the mean distance between the source andthe transfer sites increases. As explained above, a greater mean distance between the sites and thesource implies a smaller fraction of electricity being available for consumption. Accordingly, theconcave association between capital and electricity consumption reflects the difficulty in deliveringincreasing amounts of electricity to machines connected to the power grid, when the size of thenetwork expands. In essence it implies that the supply of electricity inevitably will run into‘‘diminishing returns’’ as the capital stock is expanded. This will ultimately be the reason why thephysically sustainable stock of capital per capita is bounded from above, absent technologicalprogress.

Notice that the scaling coefficient, a, should fall in a [1/2, D/(D+1)] interval. For a more preciseprior, we need to pin down D. As argued above, the most natural notion is probably that D=3. In this

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797 787

case the scaling exponent a should fall in the interval [1/2, 3/4], depending on the efficiency of thesystem.

Before we turn to dynamics some remarks on a central assumption, the existence of a unique powersource, are in order. If networks are efficient, then the theory is straightforwardly extended towardsthe existence of multiple power sources. In this case we can invoke a standard replication (constantreturns to scale) argument. Efficiency requires that each transfer site is fed by a unique source and thatthe existing sub-networks are similar (of equal size). The total network and the implied total energyflow can be obtained by simply adding up sub-networks and their energy flows. Admittedly, realelectricity-distributing networks are more complicated (see Lakervi and Holmes, 1995). In particular,they display purposeful redundancy: larger subsystems (like cities) are connected to multiple sourcesin order to avoid or limit the damage done by system failures (power cuts). In other words, realnetworks are inefficient by design. At a smaller scale of sub-networks, however, there is noredundancy. For example, usually there is just one electricity line to each house and apartment, justone cable connecting outlets with machines, and just one wire feeding any transistor in a micro-chip.We thus expect that redundancy from interconnecting lines between large-scale subsystems to haveonly mild impact on aggregate efficiency of the network.

3. A theory of capital accumulation and electricity consumption

Electricity is used to run, maintain, and create capital. Here we use a notion of capital that isbroader than the national accountants’ definition. It includes all electricity consuming appliances.That is, the k appearing here also includes durable consumption goods. As a result, we do notdistinguish whether, for example, an air-conditioning system is placed in a firm or a privatehousehold. Assume time is continuous, and let m be the energy requirement to operate and maintainthe generic capital good while n is the energy costs to create a new capital good. In that case energyconservation implies

EðtÞ ¼ mKðtÞ þ nKðtÞ: (3)

We may think of Eq. (3) as capturing demand for electricity at any given instant in time; from anaccounting perspective the right hand side of the equation summarizes the instantaneous electricityrequirements.11

Note that Eq. (3) provides a new metric for aggregation of capital. Here we measure aggregatecapital in energy-units, i.e. by the amount of electricity needed to create, use, and maintain theaggregate capital stock, While this thermodynamic measurement of capital is certainly superior to theconventional economic measurement of capital because it is not plagued by the well-knownaggregation problem (see Cohen and Harcourt, 2003 for a survey) it has also its drawbacks. Inparticular we cannot distinguish between capital in the conventional accounting sense and consumerdurables since both are connected to the same electricity network. We thus cannot (yet) integrateinvestment demand in the conventional sense into the model, a feature, which in turn preventsinferences about economic growth in terms of GDP. We can, however, derive inferences abouteconomic growth in terms capital accumulation and electricity consumption.

Inspection of (3) shows that that if we were to shut off electricity use, the capital stock would bedeclining over time at the rate m/n, due to lack of maintenance and replacement. Hence, regardless ofthe missing link to investment in the conventional sense, we observe a correspondence to the rate ofdepreciation in the conventional sense. It is given by the ratio m/n at which the capital stockdeteriorates without electricity supply.

11 One may object to the conservation equation from the perspective that human energy is also used to run, maintain and build

capital. This would call for an additional term on the left hand side, capturing metabolic energy supply by the labor force. In per

capita terms, however, the human contribution is in contemporary societies minuscule compared to the non-human energy

use. Moses and Brown (2003) puts it nicely into perspective (p. 296): ‘‘The per capita energy consumption in the United States is

11,000W . . . which is approximately 100 times the rate of biological metabolism and, . . . [it] is the estimated rate of energy

consumption of a 30,000-kg primate’’. From this perspective nothing much lost by ignoring the human physical (as opposed to

intellectual) contribution to the growth process.

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797788

Dividing through by population size P in the energy conservation Eq. (3) we get

eðtÞ ¼ mkðtÞ þ nKðtÞPðtÞ :

Assume that the population grows at a constant rate of n. Then KðtÞ=PðtÞ ¼ kðtÞ þ nkðtÞ. Inserting thisand the power law association (2) into the above equation provides the law of motion for capital:

kðtÞ ¼ en

kðtÞa � mnþ n

� �kðtÞ: (4)

The dynamical analysis is straightforward. Formally, the model shares the technical propertieswith the Solow model, where technology is assumed to be Cobb–Douglas. In particular, there exists aunique globally stable steady-state to which the economy adjusts.

Mechanically, the adjustment process works as follows. At any given instant in time k ispredetermined. Given k, a size of the underlying network is implied and consequently the supply ofelectricity (which is assumed to adjust), e, is determined. If e is sufficiently large, i.e. it exceeds theenergy needs required to maintain and run existing capital, the stock of capital can expand further.However, as the network expands the amount of additional energy which can be made available fordirect use starts to diminish (i.e. E/F declines). Eventually, therefore, the system settles down at asteady-state level of k=k�� [(m+nn)/e]1/(a�1). This can be seen as the maximum sustainable stock ofcapital, given the parameters. Below we argue that n, a and m should be given a technologicalinterpretation. Hence, if these parameters change then k� changes as well. Accordingly, the steady-state is to be viewed as a moving ‘‘target’’.

By combining the power law (1) with (3) we have in effect assumed that demand for electricityequals supply at any given instant in time. As a result we obtain a law of motion for feasible capitalaccumulation from a physical perspective. In practise, households and firms decide on how muchcapital to accumulate. Standard neoclassical growth theory provides a basis for analyzing this processof economic capital accumulation. The law of motion we have derived represents the upper contour ofwhat an economy at most can sustain if the energy conservation law is to be fulfilled and electricity isto be distributed to serve the (increasing number of) machines. In general, therefore, the aboveequation may not hold positive implications for economic accumulation and growth. For example,firms and households could, in principle. accumulate machines, in spite of the lack of an ability to‘‘feed’’ them with electricity.

The model can alternatively be expressed in terms of electricity consumption, which turns out to beuseful empirically. Formally, insert (2) in (4) to obtain the law of motion of per capita electricityconsumption.

eðtÞ ¼ ae1=a

n� eðtÞ2�1=a � a

mnþ n

� �� eðtÞ: (5)

Inspection shows that there exists a unique non-trivial steady-state of electricity consumption wheree=e��e[e/(m+nn)]a/(1�a). Since the second, negative term in (5) enters linearly, the steady-state isglobally stable if the first, positive term enters less than linearly, i.e. for 2�1/a<1. To see this formally,stability requires that

@e

@eje¼e�¼ ae1=a

n2 � 1

a

� �� ðe�Þ1�1=a � a

mnþ n

� �< 0 , 2 � 1

a

� �e� < e�:

Since 2�1/a<1 implies a<1 and for any conceivable network a2 [1/2, 3/4], the steady-state isglobally stable.12 For a sample of electricity-distributing systems the theory thus predicts b-convergence; electricity consumption per capita converges towards the same steady-state fornetworks sharing the same underlying parameters.

12 For a three dimensional network, a2 [1/2, 3/4]. For a hypothetical two-dimensional network, a2 [1/2, 2/3].

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797 789

4. Discussion

The model holds various implications on issues of whether growth can be sustained, theplausibility of neoclassical growth theory and on technological change and economic growth. Theseimplications are discussed in turn.

4.1. Neoclassical growth theory and its critics

The model allows some reconciliation between neoclassical growth theory and the work of itsstaunchest critics (Daly, 1977; Georgescu-Roegen, 1976). The central charge is that energy isintroduced into the standard models in an unsatisfactory way (if not ignored altogether). That is, byincluding energy in the aggregate production function as a separate input, which can be substitutedfor by capital. This approach is fundamentally flawed, the argument goes, because it does not considerthat any capital good is itself produced by means of energy. Here, we have explicitly taken into accountthat all capital is created, run, and maintained through energy use. Interestingly, we neverthelessarrive at a law of motion for capital which is structurally identical to that implied by the Solow model.Hence, the structure of the Solow (1956) model is not at variance with fundamental physicalprinciples, like energy conservation.

4.2. Can growth be sustained?

In the last section we found that the maximum sustainable stock of capital per capita is boundedfrom above. For a proper assessment of the steady-state result note that it is not derived from theassumption of limited supply of energy. Instead, convergence towards a constant capital stock percapita is a consequence of energy demand, distribution, and the entailed decreasing returns fromexpanding networks. It is sometimes argued that economic growth is ultimately limited from aboveby energy availability (Daly, 1977). Interestingly, however, the present analysis demonstrates that –absent technological progress – economic growth is limited even if energy supply were unlimited. Thisbrings us to the issue of how ‘‘technology’’ is said to be present in the model above.

4.3. Technological change

In the context of the model, technological change will come in the shape of changes in the keyparameters: n, m and a. That is, innovations which lower electricity requirements or improve theefficiency of the network. These innovations are critically important in facilitating capitalaccumulation. Without them, the economy would eventually end up at e� and capital accumulationcomes to a halt at k� on purely physical grounds.

Innovations that map into these parameters are easy to think of. The fact that technologicaladvances make appliances less energy-hungry because, for example, smaller sizes of computers ormicrochips are fulfilling the same task that required large, energy consuming machines in the past, iscaptured by the model as a reduction of m. A one time reduction of m leads to temporarily highergrowth according to (4) and convergence towards a permanently higher steady-state k�. A perpetuallydeclining m could be conceptualized as convergence towards ‘‘the weightless economy’’ (Quah, 1999).

A permanently lower value of n captures an innovation that admit new machines to be produced atlower energy costs. Such a parametric change seems to fit quite nicely with the sort of innovationsgrowth theories usually refer to as ‘‘general purpose technologies’’ (GPT). GPT innovations are viewedas ‘‘fundamental’’ innovations which tend to ‘‘reset’’ the economy, and instigate (ultimately) a growth‘‘spurt’’. The process, however, may involve a non-monotonous adjustment process, with an initialslump of productivity while the GPT forces a replacement of old machines with new ones that employthe new basic technology.

Bresnahan and Trajtenberg (1996) who where among those who initiated GPT research asked (p.84): ‘‘Could it be that a handful of technologies had a dramatic impact on growth over extendedperiods of time? What is it in the nature of the steam engine, the electric motor, or the silicon wafer,that make them prime suspects of having played such a role?’’ They gave a very broad answer which is

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797790

still used in the literature (see e.g., Jovanovich and Rousseau, 2005). The technology must be pervasive(spread to most sectors), there must be scope for improvement over time (lowering the costs of its use)and it must be innovation spawning, i.e. it enables the production of new products. The following‘‘handful’’ of technologies are usually referred to as GPT’s: the waterwheel, the steam engine,electricity, railways, motor vehicles, and IT.

Based on the theory developed above we can suggest a more precise answer to Bresnahan andTrajtenberg’s question. A GPT must improve either the use of energy (waterwheel, steam), its deliverythrough a network (railways, cars) or both (electricity, IT). Interestingly, while not all proposals of GPTcandidates available in the literature coincide perfectly, electricity and IT, the technologies thatrevolutionized both the use and distribution of energy, are always on the lists. Speculating about whatcould possibly be the next GPT experts usually come up with nano-technology; again a new system fordistributing energy at a new (finer) level of network with less power loss. With our theory at hand itbecomes intuitive why other seemingly equally fundamental innovations (e.g. the decoding of theDNA) are not GPT’s: they do not (much) improve the distribution and use of non-human energy.

An ad hoc way to mimic the arrival of a GPT within the standard Solow model is to simultaneouslyvary total factor productivity and the depreciation rate. The first parameter change is meant to capturethe long-run increase in productivity, and the second captures the initial slump, originating fromobsolescence of machines embodying the old technology (see Aghion and Howitt, 1998, Ch. 8.4). Theproblem is that both measures move the steady-state in opposite directions and some fine-tuning isneeded to create the desired transitional and long-run effects.

In the current model, we have a – while admittedly equally ad hoc– somewhat more elegant way toproduce the desired growth trajectory: a decrease of n. For the following discussion we assume – basedon the empirical evidence presented below – that (4) has not only bearing on physically feasiblegrowth but also on actual economic growth. A lower value for n means that machines can be producedat lower energy costs, which, for example, could have been initiated through the transistor replacingthe energy-intensive vacuum tube in electronic devices. From inspection of (4) we see that a lower nhas a double effect. It raises both the first term, ‘‘productivity’’, and the second term, ‘‘depreciation’’. Itis easy to see from the steady-state solution for k that a lower value for n unambiguously raises thesteady-state level. Starting at the original steady-state (using tube technology) k equals zero initially.Evaluating the RHS of (4) after the fall of n, we see (since a<1) that initially the negative effect throughthe depreciation channel dominates. In conclusion, energy saving technological progress causes GPT-like adjustment dynamics with an initial slump, recovery, and convergence towards a higher steady-state level.

5. Empirical evidence

In this section we confront the model with cross-sectional data for the 50 US States, covering theperiod 1960–2000. In order to do so, we must deal with two major issues: one practical, oneconceptual.

The practical problem is that data on our notion of capital per capita, k, does not exist as itencompasses both electrical appliances in homes, and in factories. Only the latter constitutes capital inthe System of National Accounts. Hence, we cannot test the model’s predictions regarding capitalaccumulation. However, we can also state the law of motion in terms of electricity consumption percapita, as demonstrated above. Electricity is measurable, which makes it feasible to perform empiricaltests.

The conceptual issue relates to whether tests are meaningful a priori. As it stands the model is onethat relates to the physically feasible amount of capital accumulation, which need not coincide withactual capital accumulation. The pragmatic way to resolve this issue is to try and see how well themodel fits the data. If it proves to be a poor fit, this is evidence that the economy is not operatinganywhere near the physical boundary.

In judging the fit of the model it is fortunate that the theory provides a set of very strongpredictions. In particular, the network theory predicts that the estimate for the scaling parameter a

should fall in an interval between 1/2 and 3/4. The model also holds structural predictions, reflected inrequired parameter restrictions, as discussed below. Finally, the model holds predictions about the

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797 791

nature of the growth process with respect to electricity consumption per capita: conditional b-convergence should prevail.13

5.1. Testing a key assumption

Before we test the model in it its entirety, we test the validity of the crucial underlying assumptionof the model, using our cross-sectional data set for US States: the postulated linear associationbetween total electricity consumption and total population (Eq. 1). This is the first place where thetheory may fail.

Our test is based on the following specification:

log ðEiÞ ¼ a0 þ a1 � log ðPiÞ þ ei; (6)

where i denotes the unit of observation. The variable ei is a noise term. That is, structurally LDi ¼ a0 þ ei,

which allows the size of the network to vary across units of observation. Our assumption, for which werequire validation, is that a1 =1. The identifying assumption for ordinary least squares (OLS) to deliveran unbiased estimate for a1 is that E(e �P)=0, where E(�) is the expectation operator. In terms of themodel, this means that the size of population is uncorrelated with the capital-labor ratio, since thelatter determines L. Under the null (the model is correct), this assumption is plausible.

We obtained data for electricity sales across US states from the US Energy InformationAdministration. Specifically, electricity sold to end users: the residential sector, the commercial sector,and the industrial sector.14 From this source we also obtained data for state populations.

Fig. 1 provides a visual impression of the results from estimating Eq. (6) by OLS and reports theparameter estimates; the regression relates to the year 2000. As is visually obvious, we cannot rejectthe assumption of proportionality between total electricity consumption and population size. Thecoefficient for log(P) is close to 1, with a fairly narrow 95% confidence interval. Hence, Fig. 1 informs usthat a potential empirical failure of the theory cannot be ascribed to a violation of Eq. (1). This evidence

Fig. 1. Population size and electricity consumption across US states. Electricity consumption vs. population size: 50 US States in

2000. The line illustrated is estimated by OLS, and yields the following result: log(Electricity)=3.95+0.98�log(Population),

R2 =0.92, 95% CI for population coefficient is (0.87,1.08). A test of the hypothesis that the slope is equal to one returns a p-value of

0.63 and thus cannot be rejected.

13 Hence, in this section we employ a basic approach to testing for convergence which goes back to Mankiw et al. (1992).

Related convergence equations have been estimated in an effort to understand energy efficiency, the ratio of total energy

consumption and GDP (e.g., Mulder and De Groot, 2007), as well as pollution (e.g., Brock and Taylor, 2004).14 Data was obtained through the web site http://www.eia.doe.gov/emeu/states/_seds.html.

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797792

corroborates previous findings for German cities and Chinese urban administrative units (Kuhnertet al., 2006; Bettencourt et al., 2007).

5.2. Testing the full theory

The second test concerns the ultimate structure of the model. In order to facilitate estimation welog-linearize (5) around the steady-state where e=e��e[e/(m+nn)]a/(1�a):

ln ½eðtÞ� ¼ ln ½eð0Þ�exp ð�ltÞ þ ð1 � exp ð�ltÞÞ 1

1 � aln ðeÞ � a

1 � aln ðnÞ � a

1 � aln

n þ mn

� �� �;

where l�(1�a)(n+m/n).This equation is, from a mechanical perspective, similar to that implied by the Solow model. The

conventional approach to taking this model to the data consists of estimating the convergenceequation by OLS (e.g., Mankiw et al., 1992).

However, the equation is non-linear in the parameter of interest (a). As a result, the model shouldnot be estimated by OLS. Instead we will have to resort to an iterative procedure. Below we thereforereport the results from estimating a by non-linear least squares (NLS).15

Unfortunately, some of the variables entering the estimation equation are unobservable: theenergy costs of running and maintaining capital (m), the costs associated with creating new capital (n),and the efficiency parameter e. From the theoretical discussion above, however, we have a prior for theratio m/n; it should reflect the rate of capital depreciation. Hence, a reasonable number for m/n can beimposed from the literature. A typical finding for the US is a rate of capital depreciation of 6% (Nadiriand Prucha, 1996; McQuinn and Whelan, 2007), which we therefore impose a priori.

In sum, the equation we estimate on the cross-section of US states is:

ln ½eiðTÞ� � ln ½eið0Þ� ¼ �½1 � exp ð�b1 � ðni þ 0:06Þ � TÞ�½b2 þ b3 ln ðni þ 0:06Þ þ ln eið0Þ� þ ui; (7)

As already noted, we test the model using cross-sectional data for the period 1960–2000; accordingly,T=40. The predictions of our theory are:

(1) b1>0,(2) b3>0 and(3) 1�b1 =b3/(1+b3)=a2 [1/2, 3/4].

If any of these three requirements are left unsupported, the theory – as it stands – is rejected. Theparameter b2 captures the influence from e and n; the model provides no specific guide in terms ofwhat to expect in terms of sign or size.

The identifying assumption needed for unbiased estimates is independence between the errorterm, ui, and the right hand side variables. In particular, this requires that if n, e and m varies across theUS states that this variation is random. We find this identifying assumption plausible. The parametersn, e and m are arguably of a technological nature, as discussed above, and the US states areundoubtedly fairly homogenous in this respect; innovations are likely to spread rapidly.

Before we turn to the NLS estimates it is worth showing the basic correlations in the data. ConsiderFig. 2 which shows the unconditional association between growth in electricity consumption percapita 1960–2000 versus initial electricity consumption per capita. The partial correlation is highlysignificant, as is obvious. Fig. 3 shows the unconditional correlation between growth in electricityconsumption per capita and the log of average state population growth 1960–2000 (plus 0.06). Inkeeping with the theory the association is negative, and significant. A simple OLS regression where thenonlinearities are ignored can account for 60 % of the variation growth of electricity sales per capitaacross the 50 states. Naturally, the OLS estimates will be biased (since the empirical model is

15 To our knowledge Dowrick (2004) is the first to make this point in the context of cross-country growth empirics, and to

estimate the (augmented) Solow model by NLS. We essentially follow his approach when estimating Eq. (6).

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797 793

misspecified) and cannot form the basis for statistical inference. The real test of the model requires usto invoke NLS.

Table 1 reports the results from estimating Eq. (7) by NLS. In panel A we report the results fromestimating the model freely. That is, without imposing the restriction 1�b1 =b3/(1+b3).

As can be seen from the R2, the model accounts well for the cross-state variation in log changes inelectricity consumption per capita over the 1960–2000 period. Moreover, the signs of the parametersare as predicted: b1>0, b3>0; both are significant at the 1% level of significance. Second, and moreremarkably, the structure of the model is not rejected by the data either. That is, the implied equality of1�b1 and b3/(1+b3) is not rejected. Consequently, we can proceed to estimate the restricted model,where a=1�b1 =b3/(1+b3) is imposed. This allows for a unique estimate for a, which we can comparewith our prior.

The results are shown in Panel B. The point estimate of the scaling exponent a is 0.62; close to 2/3.Interestingly, the confidence interval for a coincides almost perfectly with the predicted range for a, as

Fig. 2. Convergence. The figure shows the correlation between initial (log) electricity consumption per capita and subsequent

growth in electricity consumption per capita: 50 US states, 1960–2000. The straight line is estimated by OLS.

Fig. 3. Growth and state population growth. The figure shows the correlation between log state population growth (+0.06) and

growth in electricity consumption per capita: 50 US.

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797794

suggested by theory with D=3: a2 [1/2, 3/4]. In fact, we can reject a being smaller than the lowestadmissible value under the theory: 1/2. At the same time, we can reject a=3/4 at the 5% level.

In theory, it is not entirely clear what the relevant dimensionality D of the network is. However, theestimate for a suggest D=3 is probably a good approximation. If, instead, D=2, we would expect a tofall in an interval between 0.5 and 2/3. Since the latter upper boundary can be rejected by the data(which support values nearly as high as 3/4), the notion that D=2 is inconsistent with our results.

With D=3 our estimates suggests the presence of inefficiencies in the electricity distributingnetwork. As discussed in Section 2, this would be a consequence of the need for redundancy. The factthat our estimates are consistent with the fact that electricity networks, as compared with thebiological counterparts, should be (mildly) inefficient is encouraging.

In sum, the above results support a model of electricity consumption which builds on two elements:(i) log–log scaling between capital per capita and electricity consumption per capita, and, (ii) anaccounting equation relating electricity consumption to capital use, maintenance and accumulation. Thestate-level data suggests that a scaling coefficient around 2/3 is the best approximation.

The fact that the empirical model performs so well suggests that over the period in question the USeconomy must have been operating most of the time close to the feasibility constraint, as determinedby our model. This is in itself an interesting, and perhaps surprising, finding. While this appears to bethe viable conclusion, it should be stressed that our theory provides no explanation for this intriguingphenomenon. Understanding how this outcome may be brought about will require a fuller modeldetailing the intricate links between electricity consumption, investment, consumption and,naturally, GDP. Such a model has yet to be constructed.

6. Conclusion

The fundamental notion that economic growth originates from (and is limited by) energy has a longintellectual history, going back to Herbert Spencer’s (1862) First Principles. According to Spencer theevolution of societies depends on their ability to harness increasing amounts of energy for the purposeof production. Differences in stages of development can be accounted for by energy: the more energy asociety consumes the more advanced it is. Chemist and Nobel prize winner Wilhelm Ostwald (1907)developed the Spencerian ideas further. Ostwald emphasized that it is not the sheer use of energy, butthe degree of efficiency by which raw energy is made available for human purposes that defines thestage of economic (and according to Ostwald also cultural) development of society.16

Table 1NLS estimates.

Dependent variable: log change in energy consumption per capita 1960–2000

Panel A: unrestricted model

b1 0.36 (0.07)

b2 �0.38 (0.89)

b3 1.55 (0.37)

1�b1 =(b3/(1+b3)) (p-value) 0.60

R2 0.96

Obs. 50

Panel B: restricted model; 1�b1 =(b3/(1+b3)) imposed

1�b1 0.62 (0.05)

b2 �0.05 (0.88)

95% CI for a (0.52,0.73)

R2 0.96

Obs. 50

Note: Heteroskedasticity robust standard deviations in parenthesis.

16 Further refinements were made by several natural and social scientists, among them Frederick Soddy, Alfred Lottka, and

Fred Cottrell.

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797 795

The theory developed above demonstrates that this notion of development, when given a modernnetwork interpretation, is compatible with neoclassical growth theory. Indeed, it coincides with thestructural form of the economist’s core model of economic growth, the Solow growth model. Thecentral element of the theory, the network equation for electricity supply, receives support in US state-level data.

The theory has bearing on the fundamental ‘‘limits to growth’’ debate. In particular, whileconceding the importance of energy for growth, the theory also highlights the crucial importance ofhuman ingenuity. As shown above, absent technological change, growth will come to a halt even withunlimited supplies of energy, since energy dissipation increases as the economic network (appliancesand machines connected) becomes larger. This result therefore implies that technology, associatedwith the harnessing and use of energy, is as important for growth prospects as the supply of energyitself; energy and technology are equal partners in development. Indeed, as argued above, ‘‘major’’innovations (which usually are referred to as GPTs) can be seen as rare instances of progress, which ina profound way improve the harnessing, transformation, and/or distribution of energy. Integrating theliterature on endogenous technological change, with the present model of capital accumulation,would therefore seem like a useful topic for future research.

The framework could also be adapted to the study of growth in the very long run. It seems widelyconceded that human societies at large enjoy income and consumption levels of historicallyunprecedented magnitudes (e.g. Galor, 2005). A key implication of the model above is that suchincreases is inescapably linked to the ability of human societies to expand energy supply, whichrequires technological innovations. In particular then, such a long-run growth model would suggestthat the recent harnessing of electricity during the 19th century should sow the seeds of a dramaticchange in human societies. First, it is the period during which the modern day energy transportnetwork is created. That is, this period represents the genesis of e(t)=ek(t)a. Second, as a result, theseinnovations allowed for investment growth, and thus income growth, of unprecedented scale, byremoving the constraint on capital accumulation previously imposed by energy supply in ways of themetabolism of humans and animals. Accordingly, integrating the framework above with the unifiedgrowth literature also seems like a fruitful avenue for future research.

References

Aghion, P., Howitt, P., 1998. Endogenous Growth Theory. MIT Press, Cambridge, MA.Arrow, K., Kurz, M., 1970. Public Investment: The Rate of Return and Optimal Fiscal Policy. John Hopkins Press, Baltimore.Banavar, J.R., Maritan, A., Rinaldo, A., 1999. Size and form in efficient transportation networks. Nature 399, 130–132.Banavar, J.R., Damuth, J., Maritan, A., Rinaldo, A., 2002. Supply–demand balance and metabolic scaling. Proceedings of the

National Academy of Sciences of the United States of America 99, 10506–10509.Bettencourt, L.M.A., Lobo, J., Helbing, D., Kuhnert, C., West, G.B., 2007. Growth, innovation, scaling, and the pace of life in cities.

Proceedings of the National Academy of Sciences of the United States of America 104, 7301–7306.Bresnahan, T.F., Trajtenberg, M., 1996. General purpose technologies: engines of growth? Journal of Econometrics 65, 83–108.Brock, W.A., Taylor, M.S., 2004. The Green Solow Model. NBER Working Paper 10557.Cohen, A.J., Harcourt, G.C., 2003. Retrospectives: whatever happened to the Cambridge capital theory controversies? Journal of

Economic Perspectives 17 (1), 199–214.Dowrick, S., 2004. Delinearizing the Neoclassical Convergence Model. In: Dowrick, S., Pitchford, R., Turnovsky, S.J. (Eds.), E-

conomic Growth and Macroeconomic Dynamics: Recent Development in Economic Theory. Cambridge University Press,Cambridge, UK, pp. 83–94.

Daly, H., 1977. Steady-State Economics. Freeman, San Francisco.Domar, E., 1946. Capital expansion, economic growth and employment. Econometrica 14, 137–147.Galor, O., 2005. From stagnation to growth: unified growth theory. In: Aghion, P., Durlauf, S. (Eds.), Handbook of Economic

Growth, vol. 1A. Amsterdam, North-Holland.Georgescu-Roegen, N., 1976. Energy and Economic Myths. Pergamon, New York.Henderson, V., Storeygard, A., Weil, D., 2009. Measuring Growth from Outer Space. NBER Working Paper 15199.Isalgue, A., Coch, H., Serra, R., 2007. Scaling laws and the modern city. Physica A: Statistical Mechanics and its Applications 382,

643–649.Jovanovich, B., Rousseau, P.L., 2005. General Purpose Technologies, NBER Working Paper 11093.Kuhnert, C., Helbing, D., West, G., 2006. Scaling laws in urban supply networks. Physica A 363, 96–103.Kleiber, M., 1932. Body size and metabolism. Hilgardia 6, 315–332.Lakervi, E., Holmes, E.J., 1995. Electricity Distribution Network Design. Peter Peregrinus Ltd., London.Mankiw, G., Romer, D., Weil, D., 1992. A contribution to the empirics of economic growth. Quarterly Journal of Economics 107,

407–437.Maritan, A., Rigon, R., Banavar, J.R., Rinaldo, A., 2002. Network allometry. Geophysical Research Letters 29 (11) (Art. no. 1508).

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797796

McQuinn, K., Whelan, K., 2007. Solow (1956) as a model of cross-country growth dynamics. Oxford Review of Economic Policy23, 45–62.

Moses, M., Brown, J.H., 2003. Allometry of human fertility and energy use. Ecology Letters 6, 295–300.Moses, M.E., Forrest, S., Davis, A.L., Lodder, M.A., Brown, J.H., 2008. Scaling theory for information networks. Journal of the Royal

Society Interface 5, 1469–1480.Mulder, P., De Groot, H., 2007. Sectoral energy- and labour-productivity convergence. Environmental & Resource Economics 36,

85–112.Nadiri, M.I., Prucha, I.R., 1996. Estimation of the depreciation rate of physical and R and D capital in the U.S. total manufacturing

sector. Economic Inquiry 34, 43–56.Quah, D.T., 1999. The weightless economy. The Business Economist 30, 40–53.Rinaldo, A., Banavar, J.B., Maritan, A., 2006. Trees, networks, and hydrology. Water Resources Research 42 (6) Art No. W06–07.Samaniego, H., Moses, M.E., 2008. Cities as organisms: allometric scaling of urban road networks. Journal of Transport and Land

Use 1, 21–39.Solow, R., 1956. A contribution to the theory of economic growth. Quarterly Journal of Economics 70, 65–94.Stiglitz, J., 1974. Growth with exhaustible natural resources: Efficient and optimal paths. Review of Economic Studies 41, 123–

137.Suzuki, H., 1976. On the possibility of steadily growing per capita consumption in an economy with a wasting and non-

replenishable resource. Review of Economic Studies 43, 527–535.West, G.B., Brown, J.H., 2005. The origin of allometric scaling laws in biology from genomes to ecosystems: towards a

quantitative unifying theory of biological structure and organization. Journal of Experimental Biology 208, 1575–1592.West, G.B., Brown, J.H., Enquist, B.J., 1997. A general model for the origin of allometric scaling laws in biology. Science 276, 122–

126.West, G.B., Brown, J.H., Enquist, B.J., 1999. The fourth dimension of life: fractal geometry and allometric scaling of organisms.

Science 284, 1677–1679.Wilhelm, Ostwald, 1907. The modern theory of energetics. The Monist 17, 480–515.

C.-J. Dalgaard, H. Strulik / Resource and Energy Economics 33 (2011) 782–797 797